You have to exchange data (public keys and the actual encrypted content) and if you can't do that peer to peer, you need something that facilitates that. For example, webrtc clients can talk over an encrypted channel to each other p2p if they are inside the same network. If not they need a TURN server to relay that information and do NAT (firewall traversal). For obvious reasons no one wants to be the service provider of anonymous encrypted content, hence a business opportunity.
Because I see that the Keybase client is open source but not the server…